Given a source and a target probability measure supported on Rdmathbb{R}^dRd, the Monge problem aims for the most efficient way to map one distribution to the other.
This efficiency is quantified by defining a cost function between source and target data.
Such a cost is often set by default in the machine learning literature to the squared-Euclidean distance, ℓ22(x,y)=12∥x−y∥22ell^2_2(x,y)=tfrac12|x-y|_2^2ℓ22​(x,y)=21​∥x−y∥22​.
The benefits of using elastic costs, defined through a regularizer τtauτ as c(x,y)=ℓ22(x,y)+τ(x−y)c(x, y)=ell^2_2(x,y)+tau(x-y)c(x,y)=ℓ22​(x,y)+τ(x−y), was…
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